The flexibility of SAMs allow to implement and test increasingly more accurate, physically-motivated, seeding prescriptions (e.g., [1]}, [2]}) based on the specific environmental conditions required for their formation (in particular, the metals and dust content of the ISM and the level of external LW photo-dissociating radiation illuminating the halo), as discussed in Section .
As an example, Figure REF  shows the results of the statistical analysis performed by [2]}. They explore the relative birth rate of light, medium-weight and heavy seeds in a highly biased region of the Universe at \(z>6\)  by modelling the formation and evolution of a \(z=6.4\)  QSO along different analytical merger trees of the \(10^{13} M_\odot \)  host halo. In order to set the physical conditions for BH seeds formation, they implement a statistical description of the spatial fluctuations of LW photo-dissociating radiation and of metal/dust enrichment.
In this model light, medium-weight and heavy seeds form consistently with the environmental properties, i.e. the metallicity (\(Z\) , the ratio between the mass of metals and gas) and dust-to-gas mass ratio (\(\mathcal {D}\) ) of the ISM and the intensity of the illuminating LW flux (\(J_{LW}\) ).
